Two methods for spherical harmonic analysis of area mean values over equiangular blocks based on exact spherical harmonic analysis of point values

Abstract

Currently, the least-square estimation method is the mainstream method for recovering spherical harmonic coefficients from area mean values over equiangular blocks. Since the least-square estimation method involves matrix inversion, it requires great computation power when the maximum degree to be solved is large. In comparison, numerical quadrature methods are faster. Recent numerical quadrature methods designed for spherical harmonic analysis of area mean values over blocks delineated by equiangular and Gaussian grids are both fast and exact for band-limited data. However, for band-limited area mean values over an equiangular grid that has \(N\) blocks along the colatitude direction and \(2N\) blocks along the longitude direction, the maximum degree that can be recovered by using current exact numerical quadrature methods is no larger than \(N/2-1\). In this study, by using Lagrange’s method for polynomial interpolation, recently proposed numerical quadrature methods that employ the recurrence relations for the integrals of the associated Legendre’s functions are modified into two new methods. By using these methods, the maximum degree of recovered spherical harmonic coefficients is \(N-1\). The results show that these newly proposed methods are comparable in computation speed with the current numerical quadrature methods and are comparable in accuracy with the least-square estimation method for both band-limited and aliased data. Moreover, solving linear systems is not necessary for these two new methods. The error characteristics of these two new methods are quite different from those of methods that employ least-square methods. The spherical harmonic coefficients recovered using these new methods can effectively supplement those recovered using least-square methods.